EXPONENTS AND RADICALS

Explanation of topic exponents and radicals

1. Exponents and Radicals
Section 1.2

2. Objectives
Define integer exponents and exponential
notation.
Define zero and negative exponents.
Identify laws of exponents.
Write numbers using scientific notation.
Define nth roots and rational exponents.

3. Exponential Notation
a
n
= a * a * a * a…* a (where there are n
factors)
The number a is the base and n is the
exponent.

4. Zero and Negative Exponents
If a ≠ 0 is any real number and n is a
positive integer, then
a
0
= 1
a
-n
= 1/a
n

5. Laws of Exponents
a
m
a
n
= a
m+n

When multiplying two powers of the same base,
add the exponents.
a
m
/ a
n
= a
m – n
When dividing two powers of the same base,
subtract the exponents.
(a
m
)
n
= a
mn
When raising a power to a power, multiply the
exponents.

6. Laws of Exponents
(ab)
n
= a
n
b
n
When raising a product to a power, raise each factor to
the power.
(a/b)
n
= a
n
/ b
n
When raising a quotient to a power, raise both the
numerator and denominator to the power.
(a/b)
-n
= (b/a)
n
When raising a quotient to a negative power, take the
reciprocal and change the power to a positive.
a
-m
/ b
-n
= b
m
/ a
n
To simplify a negative exponent, move it to the opposite
position in the fraction. The exponent then becomes
positive.

7. Scientific Notation
Scientific Notation—shorthand way of
writing very large or very small numbers.
4 x 10
13
4 and 13 zero’s
1.66 x 10
-12
0.00000000000166

8. nth root
If n is any positive integer, then the
principal nth root of a is defined as:

If n is even, a and b must be positive.
means
nn
a b b a= =

9. Properties of nth roots





if n is odd
| | if n is even
n n n
n
n
n
mn mn
nn
nn
ab a b
a a
bb
a a
a a
a a
=
=
=
=
=

10. Rational Exponents
For any rational exponent m/n in lowest
terms, where m and n are integers and
n>0, we define:

If n is even, then we require that a ≥ 0.
/ nm n m
a a=

11. Rationalizing the Denominator
We don’t like to have radicals in the
denominator, so we must rationalize to get
rid of it.
Rationalizing the denominator is
multiplying the top and bottom of the
expression by the radical you are trying to
eliminate and then simplifying.

12. Classwork

13. Homework